The pure discrete Morse matching complex simplex-boundary homotopy conjecture

Let $cmathcal{M}_P(K)$ denote the pure complex of discrete Morse matchings of a simplicial complex $K$. Let $cmathrm{cDelta}^n$ be the $n$-simplex and $cmathrm{cpartial}cmathrm{cDelta}^n$ its boundary. Pure matching complex conjecture. For all $n cgeq 2$,

$$cmathcal{M}_P(cmathrm{cDelta}^n) csimeq cmathcal{M}_P(cmathrm{cpartial}cmathrm{cDelta}^n).$$

The equivalence holds for $n=3$, where both complexes have homotopy type $cbigvee^{81}S^3$, but the general assertion is presented as a conjectural extension and remains open.